Abstract
We have studied the radial variation of the CO abundance
in the nearby isolated globule Barnard 68 (B68).
For this purpose, B68 was mapped in the three
rotational lines
CO
,
C
O
and C
O
.
Using
the recent discovery of Alves et al. (2001) that the density
structure of B68 agrees with the prediction for a pressure bound
distribution of isothermal gas in hydrostatic equilibrium
(Bonnor-Ebert sphere), we show that the flat CO column density
distribution can be explained by molecular depletion. By combining
the physical model with the observed CO column density profile, it
was found that the density dependence of the CO depletion factor
can be well fitted with the law
,
which is
consistent with an
equilibrium between the accretion and the desorption processes.
In the cloud centre, between 0.5% and 5%
of all CO molecules are in the gas phase.
Our observations suggest a kinetic
temperature of 8 K.
In combination with the assumption
that B68 is a Bonnor-Ebert sphere, this leads to a distance of
80 pc.
The cloud mass consistent with these values is
0.7 ,
considerably less than previously
estimated. We find in B68 no clear deviance of the
near-infrared reddening efficiency of dust grains per unit H2
column density with respect to values derived in diffuse clouds.

1 Introduction

It was recently discovered by Alves et al. (2001)
that the circularly-averaged visual extinction profile of the globule Barnard 68
(B68)
agrees
remarkably well with the predictions for an isothermal sphere in
hydrostatic equilibrium or a so-called Bonnor-Ebert sphere (BES).
The suggested structural
simplicity makes B68 particularly favourable for the determination of
physical parameters and chemical abundances, and, since the cloud
probably represents a state prior to protostellar collapse, for
studying processes influencing star formation.

One important process is gas-phase depletion of molecules through
freezing out onto dust grains. CO depletion changes the chemical and
physical structure of globules by affecting the deuterium
fractionation and the ionization degree
(Caselli et al. 1998), the cooling efficiency and thermal balance,
and the gas phase chemical composition. Additionally, depletion must
be taken into account when using CO spectroscopy to trace molecular
hydrogen. Observational evidence for molecular accretion onto dust
grains comes from the detection of absorption features of molecular
ices (e.g. Tielens et al. 1991), the observation of chemical
fractionation between more and less volatile molecular species
(e.g. Zhou et al. 1989) and the comparison of column densities of
gaseous CO and dust
(e.g. Caselli et al. 1999).

The distance D to B68 cited by most authors originates from
Bok & McCarthy (1974), who allocated D=200 pc because of the
globule's
proximity to the Ophiuchus complex. However, the distance to the
Ophiuchus dark clouds was redetermined by
de Geus et al. (1989),
who found the
complex extending from
pc with a central
value of 125 pc. Other determinations of the distance to the centre of
the complex also fall in the D=125-200 pc range
(Chini 1981; Straizys 1984).
It should be emphasised that no distance
estimate for B68 itself is available, and due to the lack of
foreground stars would be difficult to obtain using classical
methods.

Alves et al. (2001) used H and K imaging of B68 to determine the
near-infrared (NIR) colours of thousands of background stars.
These extinction measurements
provided a high resolution column density profile and
yielded a physical cloud model for B68. These
achievements allow
us to study CO depletion in more detail than done before in other
objects, where both CO and H2 density distributions are
described by ad hoc models.

In the present study we determine the CO column density distribution
and the degree of CO depletion in B68 by using isotopic CO line
observations.
Combining our observations with the
extinction measurements of
Alves et al. (2001), we are able to quantify the CO abundance as a function of density and compare the observed depletion
with theoretical expectations. We also give an estimate for
the cloud's kinetic temperature.
Using this and the assumption of a BES we propose
new values for the cloud's distance, mass and
H2-to-extinction ratio.

2 B68 as Bonnor-Ebert cloud

Throughout this paper, we make extensive use of the dust column
density profile measured by Alves et al. (2001) and in parts also take up
their proposal of B68 being a Bonnor-Ebert sphere (BES).
In order to make transparent which assumptions are involved in
deriving various quantities, we summarise in this section the concept
of Bonnor-Ebert spheres and discuss its applicability to B68.

Assuming an isothermal, spherically symmetric distribution of gas in
hydrostatic
equilibrium, a cloud's density profile is governed by
Eqs. (374) and (375) of
Chandrasekhar (1939, p.156).
As Bonnor (1956) and Ebert (1955) pointed out, these equations
have a family of solutions characterised by the nondimensional radial
parameter
,
if the sphere is bound by a fixed external pressure
.
They also discovered that such a gaseous configuration is
unstable to gravitational collapse if
.
With R being the radius where the pressure P(r) has dropped
to
,
the physical parameters are determined by the scaling relation
,
where
is the central density, a the isothermal sound speed and G
the gravitational constant.

In practice, the radius of a cloud cannot be directly observed,
but only its angular diameter
and in certain circumstances its
distance D.
Using the mean molecular weight m and the central number density
to write
and
,
where
k is the
Boltzmann constant and
the kinetic temperature, we can express the
relation
between the
physical parameters of a BES as

(1)

The column density profile of a BES is determined by numerical
integration. The column density towards the centre is
,
where the constant K depends only
on
.
The shape of the density profile as well as the shape of the column
density profile depend only on
,
i.e. the normalised profiles
vs.
and
vs.
are
fully determined by
.
Vice versa, if the normalised column
density profile of a cloud is measured,
can be determined
without any knowledge of any of the parameters on the right hand side
of Eq. (1). Not even
has to be known for this,
but naturally
comes with the astrometric calibration of the
CCD image.

Alves et al. (2001) determined the extinction profile of B68 and found it
to have the same shape as the column density profile of a BES with
.
Does this mean that B68 actually is a BES? First of all, one
has to
assume that the shape of the gas column density profile has
been measured,
i.e.
.
Except for Sect. 4.1.2,
all results of this paper implicitly fall back on this assumption.
But despite having the column (and hence number) density profile of a
BES, the globule may still not be a BES in the sense that it may not
be isothermal, it may not be in hydrostatic equilibrium and it
may not be in equilibrium altogether.
It is safe to say that B68 cannot be a perfect BES for a number of
reasons: 1) Its shape is not perfectly circulary symmetric. 2) The
molecular line-widths show that a small microturbulent velocity
field is present. 3) The measured
is larger than the critical
value, i.e. B68 is unstable to gravitational collapse unless some
additional support mechanism, e.g. a magnetic field, plays a role.
Nevertheless, in the framework of inevitable idealisations in
astronomy, the BES model fits B68 certainly to some degree (as the
density profile fits near-perfectly). We will resume the BES model and
discuss the conclusion from applying it to B68 in
Sect. 6.2. Therefore we
calculate the numerical constants involved in the following
paragraphs. Note that up to and including
Sect. 6.1 we refer with
"cloud model'' and "BES profile'' to the fitted
(column) density profile, which is based on the raw observations of
Alves et al. (2001) and does not require the hydrostatic equilibrium
assumption.

The extinction profile of B68 only deviates noticeably from a BES
profile in the outermost parts
(for
,
see Fig. 2 of Alves et al. 2001). As the fitting parameters, particularly
,
change considerably if one tries to scale the BES profile to also
those data points, we have done a
-fit to the data of
Alves et al. (2001, who kindly provided us with the data), deliberately
ignoring data points at
at this stage. The lowest
was found for
,
which corresponds to a
centre-to-edge density contrast of
17.1.
The scaling parameters for
this particular BES extinction profile are
and
mag (see also Fig. 4a). The actual
peak extinction value is not needed in Eq. (1), but will be used
in Sects. 5 and 6
to derive the H2-to-extinction
ratio in B68.
Varying
by one
decimal increases
by 2%, while the scaling parameters
and
are modified by 0.6% and 0.3%
respectively. The density contrast, being an
increasing function of
,
is the least precise parameter,
changing between 16.5 and 17.7 for the given variation of
.

For a BES with
and
,
we have calculated the following values
for the radius R, the central gas particle number
density
,
the peak column
density
,
the total mass M and the external pressure
:

(2)

abbreviating
and
.
The external pressure follows from
,
as the
centre-to-edge density contrast
only depends on
.

We assume H2 and He are the only species
contributing substantially to mass and pressure, with a fractional
helium abundance of
.
Under this assumption, the mean
molecular weight is m = 2.329
(
being the atomic
hydrogen mass).
Throughout this paper, we use the commonly used rounded value
m=2.33
.
For other values of m, the quantities in
Eq. (2) have to be scaled as
,
,
and
.

If two of the three remaining unknown variables (D, T and
)
in Eq. (1) are known, the third one follows from
Eq. (1) and the cloud model is fixed.
This way we first calculated
in Eq. (2), then
from the fixed cloud model we determined
and M.
If all three variables D, T and
can be determined from
observations, Eq. (1) allows a consistency check of the
involved parameters.
In practice, for a small globule like B68
both T and
can be
determined to some accuracy e.g. by ammonia observations, while
a distance estimate remains difficult due to the
lack of foreground stars. In fact,
Eq. (1) can be used to determine the
distance under these circumstances.

The observations were carried out at the Swedish-ESO Submillimetre
Telescope (SEST) in two runs: February 1993 and May 2000. During the
first session we mapped B68 in the
CO
and C
O
lines at 110.2 and 109.8 GHz, respectively.
The map grid
spacing was 20
in the cloud centre and 40
in the
outer parts. The receiver used during these earlier observations was a
3 mm Schottky mixer dual channel receiver. By using two mixers tuned
to the same frequency we observed orthogonal polarizations
simultaneously. The system temperature,
,
reduced to
outside of the atmosphere, ranged from 300 to 400 K. In the mapping
the observing time per position was typically 2 min, which
resulted in an rms noise level of about 0.1 K. During the second
period in May 2000 we mapped the cloud in C
O
and C
O
at 109.8 and 219.6 GHz simultaneously with 3 and
1.3 mm SIS receivers. The grid spacing was
.
The system
temperatures at the two frequencies were around 160 and 260 K, and the
observing time per position was typically
1 min. The rms noise
level attained was typically 0.06 K at 3 mm and 0.1 K at 1.3 mm.

The half-power beam width (HPBW) of the antenna is 47
at 110 GHz
and 25
at 220 GHz. The pointing and focus were checked at
3-4 hour intervals towards circumstellar
SiO
maser line
sources, and the pointing accuracy was typically found to be
3
.
The map was centred on
RA 17
19
34.6,
Dec -23
46
34
(1950.0).
All observations were performed in the frequency switching mode. The
two mixers used at the same time were connected to a 2000 channel
acousto-optical spectrometer which was split in two bands of 43 MHz
each. The AOS channel width corresponds to 0.12
at 2.7 mm
and 0.06
at 1.37 mm. Further details of the SEST are
available at http://www.ls.eso.org/lasilla/Telescopes/SEST.

Calibration was done by the chopper wheel method. To convert the
observed antenna temperatures,
,
to the radiation
temperatures, ,
the former were divided by the assumed
source-beam coupling efficiences,
,
where
is the beam solid
angle of the antenna and
the normalised beam pattern. For
we adopted the main beam efficiences of the telescope
interpolated to the frequencies used, i.e. 0.71 and 0.61 at 110 and
220 GHz, respectively. Our numerical estimates for
were in fact close to these values when the source was assumed to be
disk-like with a radius of 100
.

Line area maps of the C
O transitions are presented in
Fig. 1. Characteristic for the C
O
line
intensity distribution is a steep rise at the edge and a plateau at
the centre of the globule. The C
O
line area looks patchy
with maxima at the eastern edge of the globule and the prominent
"nose'' (cmp. Fig. 3) in the southeast.

4 Estimation of the CO column densities

Two different approaches were used to estimate the CO column density
distribution across the cloud.
Firstly, we used the
traditional way to derive column densities directly from the observed
lines by assuming that the excitation temperature for each
transition is constant along the line of sight
(Sect. 4.1).
Secondly,
starting from the physical
cloud model (see Sect. 2)
we used the Monte Carlo radiative transfer program developed by
Juvela (1997)
to simulate the observed profiles from the
cloud. The physical quantities were derived by fitting the
calculated spectra to the observed ones (Sect. 4.2).
In the following, the
two methods and the results are described in detail.

4.1 Line of sight homogeneity

The assumption of a constant excitation temperature
implies that either the
cloud is homogenous (the density and the kinetic temperature are
constant) or that the cloud is isothermal and the transition in
question is thermalised. Even if
this simplistic assumption is
valid, the derivation of the excitation temperatures of the observed isotopic
CO lines involves the following difficulties: 1) The
CO/C
O abundance ratio X has been observed to change from
cloud to cloud, and may also vary from the cloud surface to its
interior parts. This depends on 13C fractionation and selective
photodissociation (see e.g. Bally & Langer 1982; Smith & Adams 1984).
2) The populations of the rotational levels of C
O may deviate from
LTE, i.e. from the situation where a single value of
describes the relative populations of all J-levels. The populations
are controlled by the collisional excitation and selective
photodissociation mechanisms, which depend on the rotational quantum
number (Warin et al. 1996).
According to Warin et al. (1996) the latter process
causes that low-lying rotational levels are thermalised or
overpopulated and higher levels are subthermally excited. In their
model calculation for a dense dark cloud with a kinetic temperature
of
K (see their Figs. 6a-c), the excitation
temperatures of both C
O
and
CO
lie
close to
,
whereas
and all higher
transitions of C
O settle between 6 and 7 K.

In the derivation of the C18O column densities we have made the
following assumptions:

1.

the C
O
and
CO
transitions have the same excitation temperature
,

2.

the
CO/C
O abundance ratio X is constant throughout
the cloud, and

3.

all the higher transitions of C
O, i.e.
C
O
have the same
excitation temperature
,
which may differ from
.

This method, based on the modelling results of Warin et al. (1996), has
been used earlier by Harjunpää (2002). The possibility that Xcan be constant within a cloud is supported by the results of
Harjunpää & Mattila (1996) and Anderson et al. (1999) in CrA, where the
CO/C
O and H
CO+/HC
O+ abundance ratios
have an almost invariable value of 10.

4.1.2 Equations and results

Figure 2:
vs.
correlation
plot. Intensities are taken from Gaussian fits to the spectra at the
velocities of the C
O peaks. The solid line is the best fit of
Eq. (3),
using a non-weighted least-squares fit
to all data points with
in both
isotopes (asterisks).
The curve implies a
CO/C
O abundance ratio X = 11.2,
and
K. The dashed line is the fit to all data
points (no SNR criterion, asterisks and crosses), yielding X = 8.2and
K. Box encompassed data points have velocity
offsets >0.3
between the lines of the two isotopes and are
not used in these fits.
For comparison, the dotted line gives the low-opacity
relation for the terrestrial isotopic ratio
X=5.5.

Figure 3:
C
O column densities (contours) overlaid on the POSS-II red
plate (image). The optical image has been smoothed to 10
resolution, and the transfer function is chosen to emphasise
qualitatively the decrease in diffuse background brightness towards the
centre of B68. The column densities are derived from our
2-transition observations of C
O, assuming a constant
excitation temperature of 8 K (see text and
Fig. 2). The angular resolution is 50
.
Only
observed positions with an uncertainty
have
been used for this map. Contours are drawn from 1.5 to
5.5
1014
in steps of
0.5
1014,
using
alternating line thicknesses. The outer contours closely
follow the optical outline of the globule. In the centre, the CO map
is rather flat, while the diffuse light suggests an even further
density increase.
This behaviour is most likely to be due to CO
depletion (see also Fig. 4).

The assumptions 1 and 2
lead to the following
formula for the radiation temperatures at a certain velocity:

(3)

where
and
are the radiation temperatures of the
CO
and C
O
lines, respectively;
is defined by

(4)

and the function
is defined by

(5)

where
is the transition frequency, h is the Planck constant,
and k is the Boltzmann constant. In Fig. 2
we have plotted the observed
CO and C
O
radiation temperatures
at the velocities of the C
O peaks. The best fit of
Eq. (3) to the data with X=11.2 and
K is
presented as a solid curve. The obtained value for
implies
that
K .
To give an impression of the accuracy of the determined parameters, we
also show a second fit (dashed line), which also uses
data points of low signal-to-noise ratio (SNR).
The value for X (now 8.2) is substantially lower, but
(now 8.4 K) differs by only 5%.

Using the derived value for
,
the excitation
temperature
can be solved from the following
equation for the observed C
O
/
integrated intensity ratio:

(6)

For this calculation, the two C
O data sets were convolved to
a common 50
Gaussian beam.
The total C
O column density was calculated by using
assumption 3.
The derived distribution of the
C
O column density is shown in
Fig. 3.
As can be seen in this figure, the column
density distribution is relatively flat. The maximum is
5.5
1014,
and the
familiar figure of the cloud is outlined by the level
3.0
1014.
In addition, the excitation temperature shows little variation in
the central part of the cloud. The average
above the column density level
2.5
1014
is 6.2 K and the
sample standard deviation is 0.6 K.

Figure 4:a) C
O column density profile of B68.
Crosses mark the same data points presented in the contour map
of Fig. 3 (i.e. 50
resolution, uncertainty
<30%). The centre of the globule is at
,
.
The solid line results from
averaging over rings at intervals of 10
.
For comparison, the
long-dashed lines give the BES
profile with
= 7.0
and angular radius 106
;
black:
convolved to our 50
resolution, grey: unconvolved.
The normalised theoretical
profile is arbitrarily scaled in y-direction with a factor of 30.3.
b) Assuming a depletion law according to
Eq. (7), the depleted and smoothed profile is given as
a black dashed line. The
parameter Z in
Eq. (7)
and the scaling in y-direction
were determined in a
-fit to match the
measured profile (solid line); individual measurements (crosses in
a)) have been omitted for visibility.
The scaling corresponds to a
no-depletion curve as given by the grey dashed line (unsmoothed).
While the dashed lines show the best-fitting depletion model
(Z=180 yields the lowest
), the other lines show profiles for
other values of Z; dash-dot: Z=20;
dash-dot-dot-dot: Z=0. In all cases, the grey lines show the
unconvolved total (gas+dust) C
O column densities.

4.1.3 Column density profile

We have plotted the radial distribution
of the C
O column density
in Fig. 4a. Also shown in this figure is the BES
column density profile
deduced from extinction measurements
(see Sect. 2 and Alves et al. 2001).
After convolving the BES column densities with a 2-dimensional
50
(full width at half maximum) Gauss function, we can exclude a constant CO
abundance in the gas phase. The difference between the measured and
the smoothed profile
strongly suggests CO depletion in the inner part of B68.

To quantify the observed depletion, we assume a depletion law of the
form

(7)

where the depletion factor
is an increasing function of
gas density; we write
for the fractional C
O abundance
and designate
as the
no-depletion fractional abundance.
Equation (7) follows from the
assumption that
the fractional CO abundance is governed by accretion onto dust grains
and desorption processes, which return the molecules into the gas
phase. The accretion rate in an isothermal cloud is
,
while the
desorption rate
can be written in
the form
.
The adsorption and
desorption coefficients A and B depend primarily
on grain
properties, temperature and the flux of heating particles, as
discussed in Sect. 6.1.
In a steady state condition the relation
holds,
and hence

(8)

Without knowing what
actually is, we can determine Z from
Fig. 4, calculating later
;
we assume
to be
5/6 (Sect. 2).

Starting from the BES density profile,
we first calculated a depleted column density
distribution for various values of Z, then smoothed the results
to our 50
resolution. Finally, we used the
-method
to scale the theoretical depleted profile to the measured profile,
which corresponds to determining
.
Figure 4b shows the profile with the lowest
,
which has Z=180 and
.
This profile reproduces the observed profile from
to
almost perfectly.
For small values of Z the fits quickly become worse, with
increasing by 15% for Z=50 and by 100% for Z=20.
Larger values of Z cannot be excluded from the
depletion analysis alone, because
increases by only a few
percent for
.
This is so because for very large Z the scaling (which determines
the gas+dust abundance) can compensate any futher increase in Z.
Only by requiring that
does not
rise into physically unrealistic regimes we can give an upper limit
for Z, which will be discussed in Sect. 6.

4.2 Monte Carlo simulations

We constructed a series of isothermal model clouds with kinetic
temperatures
between 6 and 16 K and distances between 40 and 300 pc,
all with a BES-like
density distribution (
and
).
The radiative transfer problem was
solved with Monte Carlo methods (Juvela 1997).
The radiation
field was simulated with a large number of model photons resulting from
background radiation and emission from within the cloud, where
a microturbulent velocity field was assumed.
The simulations were
used to determine the radiation field in each of the 40 spherical shells into
which the model was divided and to derive new estimates for the level
populations of the molecules. The whole procedure was iterated until the
relative change in the level populations were 10-4 between
successive iterations.

The observed C
O spectra were averaged over rings at intervals of
20
at distances 0
to 100
from the
selected centre position
RA 17
19
37.1,
Dec -23
46
59
(1950.0).
The effective
resolution of the averaged C
O
and
spectra are
60
and 30
respectively. Corresponding spectra were
calculated from the models and
averaged with a Gaussian beam to the resolution of the observations. The
correspondence between the observed spectra and the model was measured with a
value summed channel by channel over all spectra. Averaged spectra at
different distances from the centre position were all given equal weight in
the fit.

For each pair of T and D,
two parameters were optimised: the fractional C
O
abundance
and the turbulent line
width. The best fit was obtained for
K and
D=80 pc with
.
However, this solution still produces
far too little intensity at large offsets, the observed intensity at 100
being more than twice the model prediction.

We further studied a series of models assuming a
non-constant C
O fractional abundance,
,
in order to account for CO depletion in the cloud centre.
The density dependency was assumed to be
) ,
as suggested
in the homogenuous model analysis
(Sect. 4.1.3).
For the depletion parameter Z we tried a large range of values
between 4 and 104.

As in Sect. 4.1.3, we find Z not to be well
constrained, lying somewhere between 20 and 300. In contrast to our
earlier analysis, we now do find an upper limit for Z, just as we
can give a lower limit. The reason for this is that in the homogenuous
approach, high densities n, which go along with a high Z, do not
influence the excitation conditions (
being determined earlier),
while now the total density n and not only
is taken into
account.

For both Z=50 and Z=200, the best fitting values for the distance and the
temperature are 70 pc and 7 K respectively.
Some other specific combinations of Z, D and T within the ranges
Z=20-200,
D=70-120 pc and
T=6-8 K result in almost equally good fits.
The Monte Carlo simulations do not favour a
particular model, as the
values are not significantly
different. They do, however, favour the models including depletion as
compared to the Z=0 ones. For Z in the given range
the
values are
a factor of 1.6 lower than in the no-depletion models, and the spectra
are well fitted from the centre out to the edge of the cloud.
Figure 5 shows the correspondence between the model and the
observed spectra.

Figure 5:
Comparison of observed and modelled spectra.
The observed spectra (histograms) are averages over rings at distances
between 0
and 100
from the centre. The modelled spectra
(smooth lines) are generated using Monte Carlo simulations as
described in the text. The C
O
transition is shown in the
left column, C
O
in the right column. Each row stands for one
angular distance from the centre. The model behind the simulation
presented in this figure is an isothermal cloud with BES-like density
structure at 70 pc distance with a
kinetic temperature of 7 K and CO depletion according to
Eq. (7)
with Z=200.

5 Deriving H2 column densities from

In order to estimate the column density profile of B68,
Alves et al. (2001) derived the colour excesses E(H-K) of more
than a
thousand background stars behind the cloud (the underlying
assumption is that
).
Referring to the standard, i.e.
,
interstellar
reddening law of Mathis (1990),
they used the relationship
to plot the visual extinction profile. By convention,
is used
to present extinction or reddening data.
is however not the best parameter to be
converted to hydrogen column density, because the conversion factor
depends on grain properties, which can be different in different
environments (Kim & Martin 1996). We therefore use
E(H-K)
for converting the extinction data
to gas column density.
The best fitting BES extinction profile reaches
mag
(Sect. 2), hence follows
mag in
the centre of the globule.

From UV observations in the direction of diffuse clouds
Bohlin et al. (1978) determined the relation between reddening in the
optical and hydrogen column density:

(9)

This relationship (Eq. (9)),
which is frequently
used in "scaling'' the dust column densities to
,
may be
seriously in error in the case of dense clouds.
Practically no dense-cloud lines of sight were included in
the sample of Bohlin et al. (1978). Diplas & Savage (1994) re-examined the
ratio
for a larger sample of sight lines. They
found that the ratio increases to a value of
,
for sight lines involving target stars
located in dusty regions where
.
This increase is also expected for theoretical reasons, since
a depletion of small particles in dense clouds leads to a reduced
reddening efficiency (see e.g. Kim & Martin 1996).
A counter-example are two obscured lines of sight, studied
recently by the
Far Ultraviolet Spectroscopic Explorer (FUSE), i.e. HD 73882 with
(Snow et al. 2000) and HD 110432 behind the Coalsack dark
nebula with
(Rachford et al. 2001). For these two
lines of sight, the numerical factor in Eq. (9) is 5.1
(HD 73882) and 4.2 (HD 110432) instead of 5.8.
The conclusion is that there
is no evidence for a substantial increase in the H2-to-extinction
ratio towards those "translucent lines of sight'' in denser dust clouds
which can still be studied by means of ultraviolet observations.

Cardelli et al. (1989) derived
a family of extinction laws for
both diffuse and dense regions, parameterised by
the total-to-selective extinction ratio
.
Their extinction curve
for the canonical value
,
which is an average for
lines of sight penetrating the diffuse interstellar medium,
results in the colour excess ratio

(10)

In the case of dense dark clouds, where all hydrogen is in molecular
form, Eqs. (9) and (10) lead to

(11)

We apply this to B68 and obtain

(12)

in the centre of the globule,
keeping in mind that Eq. (11)
is ultimately derived from the diffuse dust sample of
Bohlin et al. (1978).

Even though there is reason to assume the extinction law
at near-infrared (NIR) wavelengths
to be independent of environment (Mathis 1990), the normalisation
of the extinction curve (with respect to hydrogen column density) may
still show a dependency. Observational evidence either favouring or
opposing the latter dependency
would help to assess the
applicability of Eq. (11) to B68, but
is still scarce.
The
ratio determined
towards Oph A(HD 147933), which is seen through an extinction
layer of
on the outskirts of the dense
Oph
cloud core, is close to the value indicated in
Eq. (11) (de Boer et al. 1986; Clayton & Mathis 1988).

Using NIR spectroscopy, H2 column densities can be probed in
much denser clouds. Lacy et al. (1994) have detected the
line
(4498 cm-1)
in absorption towards NGC2024 IRS2, and
they derive
.
By modelling the observed spectral energy distribution of
IRS2 at 1.65, 2.2, and 4.64 Ueurmnm,
Jiang et al. (1984)
have derived a colour excess of
in this direction
(using extinction curve No. 15 of van de Hulst 1957).
Furthermore, Maihara et al. (1990) have observed the
Br/Br
line ratio towards the compact H II region
surrounding IRS2. The resulting colour excess is
,
which corresponds to
(Cardelli et al. 1989).
Adopting the mean of these two colour excesses,
we end up with the ratio

(13)

This ratio is 2 times larger than the diffuse dust value
(Eq. (11)). However, the uncertainties are large,
and the lower limit of Eq. (13) is
close to the value in Eq. (11).
Moreover, because
the value in Eq. (13) is still based on a
single source
we cannot adopt it as a proven
observational result. It does suggest, however, that the
H2-to-extinction
ratio may increase in dense cloud cores (even for NIR extinction), and the use of
Eq. (11) for B68 may result in an underestimate of
.

6 Discussion

CO abundance distribution

In pure gas-phase chemistry models the CO abundance is very stable.
At gas densities over 103
it is practically constant at
all times. Therefore, any variation in the fractional CO abundance
observed in B68 is probably due to accretion and desorption
processes on dust grains. The adopted CO depletion law,
as described in Sect. 4.1.3, is consistent with
a steady state, i.e. with the situation where accretion and desorption
are in equilibrium. In the time-dependend depletion model of
Caselli et al. (2001)
this
situation corresponds to very late stages of chemical evolution.
One plausible theory of the origin of globules is that
they are remnants of dense cores of dark clouds or cometary globules
(Reipurth 1983). This scenario supports
the possibility that B68, being an aged object,
could indeed have reached chemical equilibrium.

The assumption that A/B is constant includes
the following assumptions:
1) The gas kinetic temperature is
roughly constant, 2) the dust temperature remains everywhere below the
critical temperature of CO desorption, which lies in the range
20-30 K (Léger et al. 1985; Takahashi & Williams 2000), and 3) the same
desorption
mechanisms are operating throughout the cloud. The constancy of the gas
temperature is already built in the adopted physical model (first
assumption).
The average dust temperature of B68 is 13 K, which we have
derived using ISOPHOT Serendipity Survey data
(for calibration see Hotzel et al. 2001).
Langer & Willacy (2001) claim the
detection of a dust core of
8-9 K. Both observational results support the validity of the second assumption.

As discussed in detail by Watson & Salpeter (1972),
Léger et al. (1985), Willacy & Millar (1998) and Takahashi & Williams (2000), the
desorption mechanisms operating in dense dark clouds with no star
formation are connected with cosmic rays, X-rays or
H2 formation on grains.
The main process in all three cases is impulsive whole grain or spot
heating, resulting in classical evaporation of adsorbed
molecules. Cosmic rays can also contribute to desorption via
chemical explosions, and via photo-desorption by UV photons resulting from
excitation of H2 molecules, but both processes are believed to be
effective in the low extinction regions only (Léger et al. 1985). These
processes are neglected hereafter, as is heating due to H2 formation on grains because of its relative inefficiency compared
with cosmic
ray heating (Takahashi & Williams 2000).
According to Léger et al. (1985) X-rays can heat small grains
with radii in the range 200-400 Å
more efficiently than
cosmic rays, while the heating of larger grains is assumed to be
dominated by cosmic rays.

The accretion and desorption constants are actually integrals over the
grain cross section distribution and the velocity or energy
distributions of the colliding particles. The accretion constant can
be written as
,
where
is the
total dust
grain surface area per H nucleon,
is
the average speed of the
molecules in question
(at 8 K:
)
and S is the sticking probability, which is generally assumed to be
unity in cold clouds (e.g. Sandford & Allamandola 1990). The factor of 2
comes
from the assumption that all hydrogen is in molecular form. The value
of
depends on the assumed grain size distribution, and
especially on the lower cut off of the grain radius, a-. For
example, the distribution adopted by Léger (1983), with
a-=50 Å, gives
.
On the other
hand, if one assumes that CO on grains with radii below 400 Å is
efficiently desorbed by X-rays or by other processes (i.e. effectively
no absorption on small grains), the effective
becomes
3.5
10-22.
The accretion constants Acorresponding to these -values are
2.1
10-17
and
5.3
10-18.

Based on the work of Léger et al. (1985) and Hasegawa & Herbst (1993),
Caselli et al. (2001) assumed that the desorption in L1544 is dominated
by thermal evaporation due to heating by relatively heavy cosmic rays.
The cosmic ray desorption rate for CO derived by Hasegawa & Herbst (1993),
,
is based on the Fe
nuclei flux derived by Léger et al. (1985), which is consistent with a
total H2 ionization rate of 10-17 s-1. Other assumptions
used for the indicated value of
are that 1) the average
grain radius is about 1000 Å (needed for the
calculation of the fraction of the time spent by grains in the
vicinity of 70 K), 2) the adsorption energy of CO
is 1210 K and 3) that the characteristic
adsorbate vibrational frequency of CO
is 1012 s-1.
The uncertainty of the
cosmic ray desorption rate is a good order of magnitude
(Caselli et al. 2001).

Assuming that the mentioned process
dominates the replenishment of gas-phase CO, we set
and use the accretion constant for
a-=400 Å. Then we get
,
and
the corresponding value
of the parameter
is about 100. In view of the large
uncertainty of the desorption constant, all Z values from 10
to 103 fit the model of Hasegawa & Herbst (1993).

The modelling of the parameter Zto the observed column density profile
in Sect. 4.1.3
gave a best fit for Z=180 and the fit became noticeably worse
for .
We must consider values down to Z=20 however, as the
Monte Carlo analysis in Sect. 4.2
still produced good results for this number.
The latter modelling also set the upper bound to ,
but an even
tighter limit is set
from the cosmic abundances of H, C and 18O:
With the
solar abundance [C]/[H
(Lambert 1978) and the
terrestrial isotopic ratio [16O]/[18O]=489
(Duley & Williams 1984, p.175), we get
.
For B68, with
(Eq. (12)), this means
.
This number is reached for
(see
Fig. 4b).
For our lower limit
Z=20 we find
,
corresponding to
13% of carbon nuclei being bound in CO molecules.
More recent determinations of solar and stellar carbon abundances
suggest that the average galactic
value is a
factor of 2 lower than the value of Lambert (1978) used above
(see discussion in Snow & Witt 1995). Therefore,
even though a relatively high
[C]/[H] ratio may be present in B68 (as in the sun),
it is save to exclude Z values exceeding 200.

The reasonable Z range derived
from our observations is thus 20<Z<200, corresponding to only 5%
to 0.5% of all CO molecules in the centre of B68 being in the gas
phase. The large overlap with
the above prediction deduced from the model of Hasegawa & Herbst (1993)
suggests that the degree of CO depletion can indeed be understood in
terms of accretion and cosmic ray induced desorption.

The derived range for
is considerably
higher than the commonly
quoted fractional abundance value
of Frerking et al. (1982), which is
based on a CO vs.
comparison in the Ophiuchus and Taurus
molecular cloud complexes.
This is to be expected
because they ultimately measured the gas phase CO.
To compare our results with their value, we have calculated the
gas-phase fractional abundance at the outer boundary:
(C18O, gas)/(H
/
.
This value lies between the values of Frerking et al. (1982) for
Taurus "envelopes''
(0.7
10-7) and "dense cores''
(1.7
10-7).
Considering that
drops towards the centre of B68,
we would have observed a lower fractional abundance
if we had directly used column densities, as has been
done in most other studies.
Therefore, our value is
comparable to the value for the Taurus envelopes.
Harjunpää & Mattila (1996) investigated
the molecular clouds Chamaeleon I, R Coronae Australis and the
Coalsack and
determined the
vs. E(J-K) relations,
corresponding to fractional abundances
between
0.7
10-7
(Coalsack) and
2
10-7
(Chamaeleon I).
is clearly lower in B68
than it is in the active star forming regions R Coronae Australis, Chamaeleon I and Ophiuchus.
This points towards a possible relation between the depletion degree
and the star formation activity.

6.2 Temperature and distance

As discussed in Sect. 2, the concept of
Bonnor-Ebert spheres imposes Eq. (1) on the relation
between certain observable parameters. Assuming that the state of
isothermal hydrostatic equilibrium does apply to B68, we can derive
its distance and mass from Eq. (2)
if the kinetic temperature and the central
column density are known.

There are several reasons to doubt the high kinetic
temperature of 16 K derived by Bourke et al. (1995).
The first comes from their own results.
By using this kinetic temperature and the
excitation temperature of the
(J,K)=(1,1) inversion transition of NH3,
they derive a hydrogen number density of
.
This is a good order of
magnitude lower than the value from the BES model.
According to Eq. (2) of
Ho & Townes (1983), an overestimate of the kinetic temperature leads to an
underestimate of the H2 number density, which suggests that
Bourke et al. (1995) used a too high value for
.
Secondly, our observations and Monte Carlo modelling results are in
agreement with the assumption of a nearly
homogenous
excitation temperature of
K
for
CO and C
O, which in turn is roughly equal to
.
These
results agree with the earlier observations of
Avery et al. (1987).
Moreover, temperatures derived from ammonia in other
globules without internal heating sources lie at around 10 K
(Lemme et al. 1996).
Finally, the modelling results of Zucconi et al. (2001)
for a BES with similar characteristics to B68 suggest that the dust
temperature decreases well below 10 K in the dense inner parts, which
is consistent with a low gas temperature in such an object.

After taking T=8 K as the most likely kinetic temperature, the
distance can be checked by using the formula for the central column
density
(
)
in Eq. (2) and the
canonical
ratio given in
Eq. (11).
As the NIR reddening
at the cloud center is
,
we find that the distance to
the cloud is about 70 pc. The adoption of the non-standard ratio given
in Eq. (13) would bring the cloud still
nearer, which would however be unlikely on the basis of the Monte
Carlo results (see Sect. 4.2). Therefore, it seems
reasonable to assume that the cloud is located on the near side of the
Ophiuchus complex, i.e. at a distance of 80 pc (de Geus et al. 1989).

Summarizing, T=8 K and D=80 pc are the most likely
values that are consistent with B68 being a Bonnor-Ebert sphere and
our own observations.
This is the first distance estimate ever for this globule,
which is not based on the ad hoc assumption that B68 is at the same
distance as the centre of the Ophiuchus giant molecular cloud complex.
The small distance and temperature values imply a significantly lower
mass than previously estimated:
The parameters of B68 as calculated by Eq. (2) are
,
,
and
Pa.
The derived value for the external pressure, which is needed to
contain the BES, is
not too far away from the pressure of the Loop 1
superbubble
(0.9-1.2
10-12 Pa, Breitschwerdt et al. 2000).

7 Summary

We have mapped B68 in
CO
,
C
O
and
C
O
.
Combining our observations with the extinction profile
of B68 (Alves et al. 2001), which closely
follows the one predicted
for a Bonnor-Ebert sphere, we have come to the following results:

1.

The kinetic
temperature of the gas is 8 K.

2.

The density dependence of the C
O abundance distribution proves
substantial molecular depletion in B68. The CO depletion factor
is well fitted with the law
.
The agreement between the
estimates for the ratio of the accretion constant and the depletion
constant, A/B, based on previous model predictions and derived from
our measurements, suggests that the degree of CO depletion can be
understood in terms of accretion onto dust grains and cosmic ray
induced desorption. In the centre of B68,
between 0.5% and 5%
of all CO molecules remain in the gas phase.

Based on the assumption that B68 is indeed a BES, i.e. being isothermal and in
hydrostatic equilibrium, we conclude futhermore:

3.

The most likely distance to B68 is 80 pc.

4.

The mass
of the globule is 0.7
,
which is considerably less than
estimated previously on the basis of other distance and temperature
values.

5.

The near-infrared reddening efficiency of dust grains per unit
H2 column density is close to the canonical value derived in
diffuse clouds. A clearly higher reddening efficiency
would disagree with our observations.

Acknowledgements

We thank Dr. João F. Alves for providing us with the data of the measured
extinction
profile of B68.
The Second Palomar Observatory Sky Survey (POSS-II) was made by the California Institute of
Technology with funds from the National Science Foundation, the National Geographic Society, the
Sloan Foundation, the Samuel Oschin Foundation, and the Eastman Kodak Corporation.
This project was supported by
Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)
with funds of Bundesministerium für Bildung und Forschung,
grant No. 50 QI 9801 3, and by the Academy of Finland, grant
Nos. 173727 and 174854.